Univariate Methods
Warning: in many examples the number of replications is desperately low. This is just to keep the examples simple and small. In real problems, it is much better to have more replications. Also, majority of examples are imaginary, so the conclusions drawn are sound according to the data presented, but can contradict to the reality.
Goodness of fit (2-test)
Example1: The expected Mendelian ratio in the second filial generation was 3:1. We observed 70 plants with dominant phenotype and 10 with recessive phenotype. Is there any significant difference between expected and observed ratio?
Us the the Nonparam./Distribution procedure; ask for Observed versus expected X. You will get:
Observed vs. Expected Frequencies (new.sta)
Chi-Square = 6.666667 df = 1 p < .009828
observedexpected (O-E)**2
O - E /E
C: 170.0000060.0000010.00001.666667
C: 210.0000020.00000 -10.00005.000000
Sum80.0000080.00000 0.00006.666667
Example 2: Comparison with Hardy-Weinberg equilibrium:
Observed numbers of plant of genotypes in a sample from a population were:
AA 20
Aa40
Aa10
First estimate p(A) from data: (2x20 + 40)/180 = 0.444
Expected relative frequencies are p2, 2pq, q2
Expected number of AA is 0.4442 x 90 = 17.777
Etc.
Note, df = number of categories – 1 – number of parameters estimated from the data (we estimated p) =
3 – 1 – 1 = 1
The number of df differs from that automatically provided by the program. You have to find the significance using Probability calculator in Basic Statistics.
Contingency tables
Example 3: Effect of chilling on seed germination:
Four sets of 50 seeds were stored at four temeratures for 3 months: 20 oC, 4 oC, -4 oC and –20 oC. The germination was 30%, 40%, 60% and 60%. Each seed was treated so that it can be considered independent observation. The contingency table is (enter number of cases, not percentages):
Germinated / Not germinated20 / 15 / 35
4 / 20 / 30
-4 / 30 / 20
-20 / 30 / 20
Enter data as (file chilling.sta):
CHILLINGGERMINAT FREQUE
11.0001.00015.000
21.0000.00035.000
32.0001.00020.000
42.0000.00030.000
53.0001.00030.000
63.0000.00020.000
74.0001.00030.000
84.0000.00020.000
Use Basic statistics, procedure Tables and Banners
In the panel specify using Specify tables the grouping variables (i.e. CHILLING and GERMINAT) and use FREQUE as weight. Check Pearson & M-L Chi-square and ask for Detailed two-way tables.
You will get:
Statistics: CHILLING(4) x GERMINAT(2) (chilling.sta)
Chi-square df p
Pearson Chi-square13.534df=3p=.00362
M-L Chi-square13.769df=3p=.00324
M-L is maximum likelihood Chi-square (G-test).
Other examples:
Example 4: 100 plots, 1m2 each were randomly located in a plot and the occurrence of 2 species (Cirsium and Agropyron) was observed. In 20 plots, both species were found, in 10 plots Cirsium only, in 20 plots Agropyron only, and in 50 plots none of the two species. Is the species’ occurrence independent? (Possible ecological explanations: Passive and active associations).
Example 5: 50 male and 50 female plants of a dioecious species were marked in the field at the start of vegetation season. At the end of the season it was found that 40% of male plants are still alive, whereas only 22% of female plants. Is the survival rate of male and female plant different?
Comparison of two means
Note: two independent samples can be compared eithe by the t-test for independent samples or by one way ANOVA with two categories (the results are identical). In the t-test, we can have the one-sided (one-tailed) null hypothesis. (two-tailed H0: 1=2; one-tailed H0:12 or 12). For both methods, we expect homoscedascity (variances are equal). For t-test, we have the possibility of version with separate estimates of variance for each sample. The decision about one- or two-tailed test depends on our a-priori knowledge and intention of the test and has to be done before carrying out the test. Note: It is a text-book true that for use of t-test, it is necessary that the data come from a normal distribution. Nevertheless, what is really important is that the means have normal distribution. Consequently the test is very robust when the sample-size is large (follows from Central limit theorem).
Two independent samples (Control (open) vs. treatment (filled)):
Example 6: Let’s compare the length of petals in two Ranunculus species (Ranunculus acer a R. nemorosus). Five independent observations (Should be probably more!) are available in each sample (what is random
independent observation and how to get it – relation of sample and population).
For Statistica, data can be entered in two ways:
A. Each sample is in separate variable:
Acer / Nemor5 / 7
6 / 8
4 / 9
6 / 6
5 / 8
OR
B. All the values are in one variable (length) and the other variable (species) is classification of cases (tells us, to which species the observation belongs):
species / Lengthac / 5
ac / 6
ac / 4
ac / 6
ac / 5
ne / 7
ne / 8
ne / 9
ne / 6
ne / 8
Classification variable can be also a numeric one (say, 1 instead ac and 2 instead of ne)
Use Basic statistics and t-test for independent samples. Select: A: Input file: Each variable contains data for one group or (B):Input file: One record per case (use a gouping variable).
If you are interested in one-tailed test, simply calculate P (one-tailed) = P(two-tailed)/2. (!!if the difference against null hypothesis goes in the direction of alternative hypothesis).
Example 7:
Compare weight of seeds of two species (ten independent observations available for each species).
Weghts:
Species A: 15, 16, 17, 15, 16, 14, 15, 16, 19 , 19
Species B: 14, 13, 15, 13, 16, 14, 12, 11, 13, 15
Calculate the t-test, P-value for two-tailed test, SD, SEM (explain the difference), confidence interval, plot multiple box and whisker-plot.
Two dependent samples (paired t-test)
Example 8. Five blocks (the experiment was carried out in Czech republic, so the block is called blok) were diveded in two half, one fertilized (Nitrogen - N) and other was control (H).:
Biomass values in particular plots:
Block / 1 / 2 / 3 / 4 / 5Fertilized / 23 / 25 / 36 / 19 / 22
Unfertilized / 20 / 24 / 33 / 18 / 21
Does fertilizer have any effect? (Consider one-tailed test, when we want to test whether nitrogen is a limiting factor in the plot)
The data are entered as in the previous case, i.e. one variable for fertilized and one for unfertilized plot, each block is a case. Ask for t-test for dependent samples. Results: t = 3.674235, df=4, p=0.021312
Other examples of paired observations: Comparison of bark thickness on northern and southern site of a tree: for each tree you have two values – one for southern, one for northern.
Comparison of students’ weight before and after visit at parents’ house.
Non-parametric counterparts:
t-test for independent samples: Mann-Whitney U test (in Statistica package Nonparametrics/Distrib., procedure Mann-Whitney U test – the data has to be in a form classificatory (grouping) variable and response (=dependent). Codes for groups has to be given.
Paired t-test (t-tesp for dependent samples) – Wilcoxon matched pairs test in Nonparametrics/Distrib. Package.
Response variables on ordinal scale, where non-parametric statistics is highly recommended:
Health state of a tree (on a scale from 0 – healthy tree ; 1 nearly healthy tree, …. – 5 –dead tree). Take care, when using the non-parametric test, you either test the hypothesis, that the distributions are identical (then there are no assumptions about distributions), or you test equality of means (or medians), but then you assume, the distribution shape is identical, and test, whether the distributions differ in location.
Comparison of more than two means – ANOVA
ANOVA for two groups and t-test are identical; multiple t-test is not advisable, because the probability of Type I error is in each of the t-tests, and consequently, probability of Type I error in at least one of the particular test is very high – this can lead to “statistical fishing”.
One-way ANOVA
(completely randomized design)
Example 9: Effect of soil type on plant height was tested in a pot experiment. 5 plants were grown in sandy soil, 5 plants in clay soil, and 5 plants in a peat soil. The final heights are in a table (in a way, how they should be entered for Statistica (i.e. grouping variable [= soil] and response [=height]) – file soiltype.sta:
(Note: soil type is a factor with fixed effect.)
CASE SOILHEIGHT
1s15.000
2s17.000
3s14.000
4s16.000
5s17.000
6c13.000
7c12.000
8c11.000
9c13.000
10c15.000
11p11.000
12p12.000
13p10.000
14p9.000
15p10.000
Use the ANOVA/MANOVA procedure.
In startup panel:
Independent (factors): soil
Dependent: Height
Press OK, and in the next panel ask for All effects
You will get the ANOVA result table:
Summary of all Effects; design: (soiltype.sta)1-SOIL
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 2 / 36.6 / 12 / 1.733333 / 21.11539 / 0.000117
As p=0.000117, we can conclude that the effect of soil type is highly significant.
Reasonable graphical presentation can be obtained by selecting: Descriptive stats & graphs, Categorized box & whisker:
For multiple comparisons ask Post hoc comparisons (unless you have a priori planned ones). Tukey is recommended.
Other examples:
Random factor (note that for the one-way ANOVA, the results are the same for fixed and random factors): Individuals from three clones of Festuca rubra were vegetatively propagated under identical conditions. Then, 5 tillers from each clone were grown, each in a separate pot, for 5 weeks and the number of tillers was calculated to find, whether there is effect of genetic variability (i.e. the difference between clones) on tillering. Results (number of additional tillers from each of original 5 tillers):
Clone 1: 6,4,5,8,6
Clone 2: 2,3,2,4,3
Clone 3: 4,6,5,7,4
Probably, the multiple comparison is meaningless.
Probably, the square-root transformation can be useful.
Non-parametric counterpart: Kruskal-Wallis ANOVA (or median test). Use procedure Nonparametrics/Distrib., Kruskal-Wallis. Panel is similar to parametric test.
When to use the log-transformation? When the data are log-normal, sd is linearly dependent on mean and effects are multiplicative.
Two-way analysis of variance: factorial experimental design
Example10:
Effect of nitrogen and watering on plant height was studied in a pot experiment. Two levels of each factor were applied (normal – 0, increased – 1)
Enter each of independent factors into one variable (file fertwate.sta)
Nitrog Water Height
10.0000.00023.000
20.0000.00025.000
30.0000.00024.000
40.0000.00026.000
50.0000.00019.000
60.0001.00032.000
70.0001.00037.000
80.0001.00034.000
90.0001.00035.000
100.0001.00036.000
111.0000.00029.000
121.0000.00028.000
131.0000.00029.000
141.0000.00031.000
151.0000.00030.000
161.0001.00057.000
171.0001.00059.000
181.0001.00062.000
191.0001.00058.000
201.0001.00059.000
Use a similar procedure as before: Nitrog and Water are independent, Height is dependent. After All effects you will get:
Summary of all Effects; design: (fertwate.sta)
1-NITROG, 2-WATER
df MS df MS
Effect Effect Error Error F p-level
111140.050163.950000288.6202.000000
212101.250163.950000531.9620.000000
121414.050163.950000104.8228.000000
Meaning of interaction: the main effect are not additive; see the picture obtained form Means/graphs after asking for interactions:
The lines are not parallel => effects are not additive.
Non-replicated BACI (Before After Control Impact)
Before:
C I
After:
C I
The response (e.g. content of Cd and Pb in algae, file noBACI.sta) is analyzed by two way analysis of variance. Main factors are WHEN (Before and After impact) and WHERE (above [Control plot] and below [Impact plot] the oil spill). The significant interaction is (with caution because of pseudoreplication) considered to be a proof of impact:
Data:
WHERE WHEN CD PB
1CB5.0004.000
2CB4.0006.000
3CB6.0005.000
4CB5.0003.000
5IB8.0006.000
6IB9.0005.000
7IB6.0007.000
8IB8.0007.000
9CA6.0004.000
10CA7.0007.000
11CA9.0007.000
12CA8.0006.000
13IA10.00011.000
14IA11.00013.000
15IA9.00012.000
16IA10.00014.000
Results:
Cd:
Summary of all Effects; design: (nobaci.sta)1-WHERE, 2-WHEN
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 1 / 27.5625 / 12 / 1.145833 / 24.05455 / 0.000363
2 / 1 / 22.5625 / 12 / 1.145833 / 19.69091 / 0.00081
12 / 1 / 0.0625 / 12 / 1.145833 / 0.054545 / 0.819271
Pb:
Summary of all Effects; design: (nobaci.sta)1-WHERE, 2-WHEN
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 1 / 68.0625 / 12 / 1.5625 / 43.56 / 2.54E-05
2 / 1 / 60.0625 / 12 / 1.5625 / 38.44 / 4.59E-05
12 / 1 / 22.5625 / 12 / 1.5625 / 14.44 / 0.00253
We have no reason to expect the effect on Cd (interaction is non-significant – accordingly, lines in graph are parallel), even when both main effects are significant. On the contrary, there is effect on Pb.
Experimental design:
Completely randomized (correct)
Randomized complete blocks (correct):
E N V I R O N M E N T A L G R A D I E N T
Block 1Block 2Block 3Block 4
Latin square design (correct)
FALSE (Pseudoreplications!!!!)
Randomized complete blocks: (Example 11: file seedlenv.sta): In an experiment set in 4 randomized complete blocks, following treatments were used: control (1), litter removal (2), Nardus removal (3) and litter and moss removal (4).
TREATMEN / BLOCK / SEEDLSUMrel1 / 1 / 1 / 95
rel2 / 2 / 1 / 91
rel3 / 3 / 1 / 64
rel4 / 4 / 1 / 107
rel5 / 1 / 2 / 88
rel6 / 2 / 2 / 70
rel7 / 3 / 2 / 51
rel8 / 4 / 2 / 180
rel9 / 1 / 3 / 44
rel10 / 2 / 3 / 57
rel11 / 3 / 3 / 55
rel12 / 4 / 3 / 173
rel13 / 1 / 4 / 94
rel14 / 2 / 4 / 99
rel15 / 3 / 4 / 53
rel16 / 4 / 4 / 80
Analyzed by two way ANOVA, (TREATMENT and BLOCK are main effect, interaction term is used as error term – of course, interaction cannot be tested)
In Statistica: use Pooled effect/error term for defining error term. You will get.
Summary of all Effects; design: (seedlenv.sta)1-TREATMEN, 2-BLOCK
Customized Error Term
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 3 / 4513.229 / 9 / 1068.84 / 4.222548 / 0.040278
2 / 3 / 215.5625 / 9 / 1068.84 / 0.201679 / 0.892645
12
Will be done automatically, when you declare BLOCK as random factor.(However, in this case, you will not get the test of block significance).
If blocks do not differ among themselves, then block structure decreases the power of the test. In example above, the completely randomized design would yield:
Summary of all Effects; design: (seedlenv.sta)1-TREATMEN
df / MS / Df / MS
Effect / Effect / Error / Error / F / p-level
1 / 3 / 4513.229 / 12 / 855.5208 / 5.275417 / 0.014964
Non-parametric counterpart: Friedman test (in Nonparametrics/Distrib.): each block is a row, each column is a treatment. In this arrangement, the parametric ANOVA can also be calculated: specify no independent variable, all the columns are dependent variables and specify the Repeated measure (within SS) design.
Example12 (file stomata.sta):
Stomatal densities on leaves, stem and petals were compared. 10 plants were used and for each plant, we have one value for leaves, one value for stem and one value for petals:
Plant / Leaves / Stem / petals1 / 9 / 6 / 7
2 / 15 / 9 / 10
3 / 7 / 3 / 4
4 / 15 / 10 / 12
5 / 11 / 7 / 9
6 / 20 / 15 / 17
7 / 19 / 18 / 18
8 / 4 / 3 / 3
9 / 16 / 11 / 13
10 / 14 / 10 / 11
Fixed and random effects
Example 13 (file ferlocal.sta): At three meadow localities, 5 control plots and 5 fertilized plots were established. The biomass at the end of the season was harvested, oven dried and weighted. Following results were obtained:
LOCALITY / FERTIL / BIOMASS1 / 0 / 510
1 / 0 / 520
1 / 0 / 525
1 / 0 / 545
1 / 0 / 500
1 / 1 / 600
1 / 1 / 610
1 / 1 / 620
1 / 1 / 610
1 / 1 / 605
2 / 0 / 400
2 / 0 / 420
2 / 0 / 410
2 / 0 / 405
2 / 0 / 430
2 / 1 / 520
2 / 1 / 570
2 / 1 / 560
2 / 1 / 520
2 / 1 / 550
3 / 0 / 680
3 / 0 / 670
3 / 0 / 650
3 / 0 / 660
3 / 0 / 670
3 / 1 / 670
3 / 1 / 650
3 / 1 / 630
3 / 1 / 645
3 / 1 / 670
Are there differences among localities? Is there any effect of fertilization? Is the fertilization effect the same at all the localities?
Compare the results when locality is a fixed effect factor:
Summary of all Effects; design: (ferlocal.sta)1-LOCALITY, 2-FERTIL
Df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 2 / 81970 / 24 / 240.4167 / 340.9497 / 2.39E-18
2 / 1 / 35707.5 / 24 / 240.4167 / 148.5234 / 9.07E-12
12 / 2 / 13710 / 24 / 240.4167 / 57.026 / 7.62E-10
And when locality is a random effect factor:
Summary of all Effects; design: (ferlocal.sta)1-LOCALITY, 2-FERTIL
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 2 / 81970 / 24 / 240.4167 / 340.9497 / 2.39E-18
2 / 1 / 35707.5 / 2 / 13710 / 2.604486 / 0.247909
12 / 2 / 13710 / 24 / 240.4167 / 57.026 / 7.62E-10
The results for the fixed factor differ considerably (the results for the other two terms are identical). There is difference in the meaning: when locality is a fixed factor, the results are to be generalized to the three localities only (i.e., on average, the fertilization increases biomass on the three localities). When the locality is a random factor, then the three localities are random sample from (potentially infinite) set of all possible localities; in this case we do not have enough evidence to say anything about the fertilization effect in the whole set (except that the effect is not the same in all the localities (significant interaction).
Hierarchical (nested) designs
Simple hierarchy: Example 14: We study the effect of soil type on seed weight. We have four pots with sand and four pots with clay. From each plant, we weighted 3 seeds. The design was:
The data should be entered as follows (file seedhier.sta):
SOIL / POT / SEEDWEIG1 / s / 1 / 6
2 / s / 1 / 7
3 / s / 1 / 6
4 / s / 2 / 5
5 / s / 2 / 6
6 / s / 2 / 5
7 / s / 3 / 7
8 / s / 3 / 7
9 / s / 3 / 6
10 / s / 4 / 5
11 / s / 4 / 5
12 / s / 4 / 6
13 / c / 5 / 8
14 / c / 5 / 7
15 / c / 5 / 8
16 / c / 6 / 7
17 / c / 6 / 7
18 / c / 6 / 8
19 / c / 7 / 8
20 / c / 7 / 7
21 / c / 7 / 8
22 / c / 8 / 6
23 / c / 8 / 6
24 / c / 8 / 6
The analysis of variance has to reflect the hierarchical nature of the design: in particular, pot (a random factor) is nested the factor soil. So in the panel, the independent variables are soil and pot, and you have first select codes for the factors (use all), this will enable to state that pot is nested within soil with 4 levels, and finaly, you have to state that pot is a factor with random effect. You will get:
Summary of all Effects; design: (seedhier.sta)1-SOIL, 2-POT
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 1 / 9.375 / 6 / 1.652778 / 5.672269 / 0.054645
2 / 6 / 1.652778 / 16 / 0.291667 / 5.666667 / 0.002538
12
It follows that (at α=0.05) we were not able to reject the null hypothesis that soil has no effect, but there is significant effect of the pot. Note, that for soil we have used as an error term MS for pot, not the residual MS. For testing the effect of soil, particular pots are the independent observations. The pots are tested against the residual (i.e. between seed within a pot) variability.
If we use (erroneously) the particular seeds as independent observations, we would get nicely significant differences between soil type:
Summary of all Effects; design: (seedhier.sta)1-SOIL
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 1 / 9.375 / 22 / 0.662879 / 14.14286 / 0.001079
Unfortunately, this is false analysis, and tremendously underestimates the Type I error probability.
Split-plot design
Split-plot is sometimes called also the simple hierarchy described above; here we will call split-plot the situation where there is a within-plot factor, effect of which is also tested.
Example 15:
The effect of fertilization was studied on 6 plots, 3 of them on limestone , and 3 of them on granit. In each field following treatment were established: control ( C ), fertilized by Nitrogen (N) and fertilized by Phosphorus (P). The design looked like:
Plot 1 Plot 2 Plot 3
Plot 4 Plot 5 Plot 6
The response was total biomass in a plot. We are interested in following questions: Is there any difference between biomass on granit and limestone (test rock), is there any general effect of fertilization (test fertil), and the effect of fertilization the same on granit and on limestone (test interaction rock x fertil). Because of the hierarchical structure, we are not allowed to use the two-way analysis of variance, but we have to include the plot (1 to 6) as another factor, which is nested within rock.
The data should be entered as (file rockfert.sta):
ROCK / FERTIL / PLOT / BIOMASS1 / g / C / 1 / 625
2 / g / N / 1 / 688
3 / g / P / 1 / 645
4 / l / C / 2 / 455
5 / l / N / 2 / 482
6 / l / P / 2 / 520
7 / g / C / 3 / 695
8 / g / N / 3 / 756
9 / g / P / 3 / 740
10 / l / C / 4 / 420
11 / l / N / 4 / 460
12 / l / P / 4 / 499
13 / g / C / 5 / 460
14 / g / N / 5 / 488
15 / g / P / 5 / 456
16 / l / C / 6 / 520
17 / l / N / 6 / 590
18 / l / P / 6 / 650
The independent variables are ROCK, FERTIL and PLOT, dependent is BIOMASS. DO not forget sssto state all the code for independent variables. Than state that PLOT is nested within ROCK (with 3 levels) and PLOT is a random factor. The final results are:
Summary of all Effects; design: (rockfert.sta)1-ROCK, 2-FERTIL, 3-PLOT
df / MS / df / MS
Effect / Effect / Error / Error / F / p-level
1 / 1 / 50880.5 / 4 / 33989.67 / 1.49694 / 0.288287
2 / 2 / 5496.167 / 8 / 248.5 / 22.11737 / 0.00055
3 / 4 / 33989.67 / 0 / 0
12 / 2 / 2710.5 / 8 / 248.5 / 10.90744 / 0.005184
13
23 / 8 / 248.5 / 0 / 0
123
Note, that for the effect of ROCK (“main plot effect)”, the PLOT MS is used as error in F calculation. We can conclude that on average, the biomass do not differ between limestone and granit, that the fertilization has a significant effect, and that the effect of fertilization is NOT the same on granit and limestone: this can be illustrated by a picture (use means/graph and plot interaction ROCK and FERTIL):
On limestone, the effect of phosphorus is higher than that of nitrogen, on granit, the reverse is true.
Replicated BACI – Repeated measurement (Example 16)
T0 Treatment T1 T2